I'm self studying Axler's LADR and, in doing a proof in the exercises, I couldn't fully justify to myself a particular step that I had to use. The question is as follows:
Now my proof went something like: Consider $v\neq0\in U$ and $w\notin U$ (but both $\in V$). Now $v+w\notin U$. Thus $T(v+w)=0 \neq Tv+Tw=Sv+0=Sv$. QED.
Now the proof rests on my assertion that $v+w\notin U$, and that makes intuitive sense seeing as $w$ is not in $U$. But I couldn't think of how to show that? Obviously, we could take $v=0$ and then this is trivially true, but I think I had to specify $v\neq 0$ so that the required inequality occurs in the last line of the proof (ie. we would get $0=0$ if we chose $v=0$). I'm sure I'm missing something easy but if someone could convince me that there exists $v$ such that $v+w\notin U$ it would be much appreciated.